Limit of product diagrams Suppose we are in a "familiar" category, like sets, groups, or topological spaces.
Consider the diagram:
$$
X\times X \overset{p_1}{\underset{p_2}{\rightrightarrows}} X
$$
where $p_1,p_2$ are the projections on the first and second component respectively. 

Is the equalizer of this diagram the object $X$, with the diagonal map?

Now, if we extend the diagram by adding:
$$
X\times X\times X \substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow} X\times X \overset{p_2}{\underset{p_1}{\rightrightarrows}} X
$$
where the three arrows "forget" one component each.

Is the limit of this diagram still the object $X$, with the (threewise) diagonal map?

 A: The universal property of products states that given two arrows $f : A \to B$ and $g : A \to C$ there exists a unique arrow $\langle f,g \rangle : A \to B\times C$ such that $\pi_1 \circ \langle f,g \rangle = f$ and $\pi_2 \circ \langle f,g \rangle = g$ and this is natural in $A$.  Spelled out, naturality states that $\langle f,g \rangle \circ h = \langle f\circ h,g\circ h \rangle$.
It's trivial then, to show that $\langle id, id \rangle : X\to X\times X$ equalizes $\pi_1$ and $\pi_2$. It's also easy to show that given any arrow $h : Y \to X\times X$ that equalizes $\pi_1$ and $\pi_2$ there exists a unique arrow to $\iota_h : Y \to X$ such that $h = \langle id,id \rangle \circ \iota_h = \langle \iota_h,\iota_h \rangle$, namely $\pi_1 \circ h = \iota_h = \pi_2 \circ h$.  This argument holds in any category where $X\times X$ is a product (we don't need any other products).
You can use the same approach to show that $\langle id,id,id \rangle : X \to X\times X \times X$ is an equalizer in your second case.  But let's step back.  A key fact is $$\text{Lim}(\lambda X.\text{Hom}(-, DX)) \cong \text{Hom}(-,\text{Lim}D)$$ whenever the limit on the right exists.  In fact, this can be taken as the definition of limit.  Since $\mathbf{Set}$ is complete, i.e. has all (small) limits, limits in a functor category into $\mathbf{Set}$ are evaluated pointwise, so for each object $Y$ we have: $$\text{Lim}(\text{Hom}(Y,D-)) \cong \text{Hom}(Y,\text{Lim}D)$$
So your second example has as components the limits of the following diagram in $\mathbf{Set}$ for each $Y$: $$\text{Hom}(Y,X)\times\text{Hom}(Y,X)\times\text{Hom}(Y,X)\substack{\to\\\to\\\to}\text{Hom}(Y,X)\times\text{Hom}(Y,X)\rightrightarrows\text{Hom}(Y,X)$$
Limits in $\mathbf{Set}$ are easy though.  They are always equationally defined subsets of a, perhaps infinite, product.  In this case we have, $$\{(x_1, x_2, x_3)\mid x_1 = x_2 \land x_2 = x_3 \land x_1 = x_3\} = \{(x,x,x)\mid x\in\text{Hom}(Y,X)\}\cong\text{Hom}(Y,X)$$ where $x_i\in\text{Hom}(Y,X)$.  Via Yoneda, this shows that $\text{Lim}D \cong X$ where $D$ is the functor representing the above diagram.
