When are canonical maps between limits monomorphisms? If $\mathbf{D}_1 \hookrightarrow \mathbf{D}_2$ is an inclusion of diagrams in a category $\mathbf{C}$, and $\mathbf{C}$ has $\varprojlim \mathbf{D}_1$ and $\varprojlim \mathbf{D}_2$, then the inclusion induces a canonical map $\varprojlim \mathbf{D}_2 \to \varprojlim \mathbf{D}_1$ of cones to $\mathbf{D}_1$.
In $\mathbf{Set}$ (I'm not sure if this is always true elsewhere, even in other toposes), taking $\mathbf{D}_2$ as $$\require{AMScd} \begin{CD} @. X \\
@. @VfVV \\
Y @>g>> Z\end{CD}$$ and obtaining $\mathbf{D}_1$ by forgetting $Z$ induces a monomorphism $X \times_Z Y \hookrightarrow X \times Y.$
However, taking $\mathbf{D}_2$ to be $X \overset{\operatorname{id}}{\to} X$ and $\mathbf{D}_1$ to be the empty diagram induces terminal maps $X \to \mathbf{1}$ for each $X$, which (at least in $\mathbf{Set}$) are almost never monomorphisms.
Going back to (a complete, say) $\mathbf{C}$, what are conditions on $\mathbf{C}$, $\mathbf{D}_2$, and $\mathbf{D}_1$ that ensure $\varprojlim \mathbf{D}_2 \to \varprojlim \mathbf{D}_1$ is a monomorphism?
 A: Let $F : \mathcal{D}_1 \to \mathcal{D}_2$ be a functor where $\mathcal{D}_1$ and $\mathcal{D}_2$ are small categories. The following are equivalent:


*

*For every locally small category $\mathcal{C}$ and every diagram $X : \mathcal{D}_2 \to \mathcal{C}$, if $\varprojlim_{\mathcal{D}_2} X$ and $\varprojlim_{\mathcal{D}_1} X F$ exist, then the comparison $\varprojlim_{\mathcal{D}_2} X \to \varprojlim_{\mathcal{D}_1} X F$ is a monomorphism.

*For every diagram $X : \mathcal{D}_2 \to \mathbf{Set}$, the comparison $\varprojlim_{\mathcal{D}_2} X \to \varprojlim_{\mathcal{D}_1} X F$ is injective.

*For every object $d$ in $\mathcal{D}_2$, the comma category $(F \downarrow d)$ is inhabited.


The equivalence of (1) and (2) is essentially a Yoneda-type argument.
To see that (1) implies (3), consider the case where $\mathcal{C} = [\mathcal{D}_2, \mathbf{Set}]^\mathrm{op}$ and $X : \mathcal{D}_2 \to \mathcal{C}$ is the Yoneda embedding. The assumption says that the comparison $\varinjlim_{\mathcal{D}_1^\mathrm{op}} \mathcal{D}_2 (F -, d) \to \varinjlim_{\mathcal{D}_2^\mathrm{op}} \mathcal{D}_2 (-, d)$ is surjective for every object $d$ in $\mathcal{D}_2$; but $\varinjlim_{\mathcal{D}_2^\mathrm{op}} \mathcal{D}_2 (-, d) \cong 1$, so this says that there exist an object $d'$ in $\mathcal{D}_1$ and a morphism $F d' \to d$ in $\mathcal{D}_2$.
That (3) implies (2) is a universality argument (because $[\mathcal{D}_2, \mathbf{Set}]^\mathrm{op}$ is the complete category freely generated by a diagram of shape $\mathcal{D}_2$), but there is also a fairly obvious direct argument.
