$\operatorname{lim} yd\cong y(\operatorname {lim}d)$ Let $\bf I$ be a small category. For a functor  $g:\mathbf I\to \mathbf{Set}$, set $\operatorname {lim} g:=\operatorname{Nat}(t,g)$, where $t$ is the constant functor to the singleton. Clearly this definition contains the same data of the definition via cones or via equalizer of the product.
Let $d:\mathbf I\to \mathbf X$ be a functor. The objectwise limit functor, $\operatorname{lim} \mathbf X(-,d):\mathbf X^{\text{op}}\to \bf Set$, on any$x\in\operatorname{ob}(\mathbf X)$ is defined as $\operatorname{lim} \mathbf X(x,d)$; hence there are canonical maps $\operatorname{lim} \mathbf X(x,d)\to \mathbf X(x,d_i)$, for all  $i$, that we call $x^i$. If $f\in \mathbf X(x',x)$, the map $\operatorname{lim} \mathbf X(f,d)$ is then the unique one satisfying $(x')^i\circ\operatorname{lim} \mathbf X(f,d)=\mathbf X(f,d_i)\circ x^i$. Notice that $\mathbf X(x,d)$ is a diagram in $\bf Set$, so the objectwise limit functor is  defined relying only on the definition given in the previous paragraph. The aim now  is to show that $\operatorname{lim} \mathbf X(-,d)$ is naturally isomorphic  to the functor $c:\mathbf X^{\text{op}}\to \bf Set$, such that $c(x)$ is the set of the cones over $d$ with vertex $x$, extended functorially in the obvious way.
Any $u\in \operatorname {lim}\mathbf X(x,d)$ yields a cone with vertex $x$, taking all the $x^i(u)\in \mathbf X(x,d_i)$: in fact since $\operatorname {lim}\mathbf X(x,d)$ is   a cone over $\mathbf X(x,d)$, for every $h\in\mathbf I(h_0,h_1)$, holds $\mathbf X(x,d_h)(x^{h_0}(u))=x^{h_0}(u)$; equivalently $d_h\circ x^{h_0}(u)=x^{h_1}(u)$. Hence we obtained a map $\phi_x:\operatorname {lim}\mathbf X(x,d)\to c(x)$.

*

*If $a,b\in \operatorname {lim}\mathbf X(x,d)$ are such that $x^i(a)=x^i(b)$ for all  $i$, call $a^*:\{*\}\to \operatorname {lim}\mathbf X(x,d)$ the map  with image $a$, and define analogously $b^*$; then the cones identified by all the $x^i\circ a^*$ or the $x^i\circ b^*$ are the same one,  implying $a^*=b^*$ (injectivity).


*Any $z\in c(x)$ consists of $\{z_i\in\mathbf X(x,d_i)\}_i$ satisfying, for all  $h$ as above, $d_h\circ z_{h_0}= z_{h_1}$, i.e. $\mathbf X(x,d_h)( z_{h_0})= z_{h_1}$. Hence the maps $z_i^*$ give raise to a cone over $\mathbf X(x,d)$, meaning that they factors through a map $\{*\}\to \operatorname {lim}\mathbf X(x,d)$ (surjectivity).
It is left to prove  the naturality of $\phi$, so let $f\in \mathbf X(x',x)$ as before. $\require{AMScd}$
$$\begin{CD}
\operatorname {lim}\mathbf X(x,d)@>>{\operatorname {lim}\mathbf X(f,d)}> \operatorname {lim}\mathbf X(x',d)\\
@VV{\phi_x}V @VV{\phi_{x'}}V \\
c(x) @>>{c(f)}> c(x')\\
\end{CD}$$ From one route, starting with $u\in \operatorname {lim}\mathbf X(x,d)$, we get to $c(f)\{(x^i(u))\}_i=\{x^i(u)\circ f\}_i$; from the other we also get $\{(x')^i(\operatorname {lim}\mathbf X(f,d)(u))\}_i=\{\mathbf X(f,d_i)\circ x^i(u)\}_i=\{x^i(u)\circ f\}_i$.
It is straightforward that being the universal  element for a representation of $c$ is equivalent to  being a terminal  cone over $d$; since $c\cong \operatorname {lim}\mathbf X(-,d)$, we see that (1) being the universal  element for a representation of $\operatorname {lim}\mathbf X(-,d)$ and (2) being a terminal  cone over $d$ are equivalent properties, that indeed characterize $\operatorname{lim}d$ (by definition).
The core of my confusion was at this point; let the Yoneda embedding be $y:\mathbf X\to \operatorname{PSh}(\mathbf X)$. If I understood, (1) amounts to saying that $y(x):\mathbf X\to \bf Set$ preserves limits for any $x$; indeed $x^i$, translated (via representing isomorphism) to a map $\mathbf X(x,\operatorname{lim}d)\to\mathbf X(x,d_i)$, is equal to $\mathbf X(x,p_i)$, where the $p_i\in  X(\operatorname {lim}d,d_i)$ are canonical. Knowing that (3) the limit of a diagram of presheaves is computed objectwise,  in the case of $y$ holds the following (the first natural bijection holds by (3), the second by (1)), whose meaning is that $y$ itself is a functor  that preserves limits. $$(\operatorname {lim} yd)(-)\cong \operatorname {lim} (y(-)d)\cong y(\operatorname{lim} d)$$
 A: *

*the yoneda embedding functor
$$ y : {\bf C} \to [{\bf C}^{op},{\bf Set}]$$
preserves all limits that exist in $\bf C$.

*limits in $[{\bf C}^{op},{\bf Set}]$ are computed "pointwise", in the sense that the limit of a diagram $D : {\bf J} \to [{\bf C}^{op},{\bf Set}]$ can be computed as the functor that sends $C\mapsto \lim_i D_i(C)$.

These two results are certainly related, but they are not "the same". The relation is that you need the second (a certain construction done in $[{\bf C}^{op},{\bf Set}]$ has the correct universal property) to prove the first.
If you know how limits are computed in $[{\bf C}^{op},{\bf Set}]$, to prove 1) it is enough to show that for each diagram $D : {\bf I}\to {\bf C}$, the functor ${\bf C}(-,\lim D_i)$ and the functor $\lim{\bf C}(-,D_i) $ are canonically isomorphic. The second functor is defined exactly as above: an object $X\in\bf C$ goes to the limit, computed in $\bf Set$, of the diagram ${\bf I} \to {\bf Set}$ sending $i\mapsto \lim{\bf C}(X,D_i)$.
Now, fix an object $X$ as above: apply the functor ${\bf C}(X,-)$ to the terminal cone $\{\pi_i : \lim D_i\to D_j\mid j\in {\bf I}\}$. You get maps
$$ {\bf C}(X,\lim D_i) \xrightarrow{{\bf C}(X, \pi_i)} {\bf C}(X,D_i).$$
This is a cone: from here you get a unique map
$$ {\bf C}(X,\lim D_i) \xrightarrow{\phantom{{\bf C}(X, \pi_i)}} \lim{\bf C}(X,D_i).$$
which is a bijection (some hard work left on you :-)). (The last bit to check is prove this iso is natural in $X$; this is less tedious than the previous point.)
