When I work with various presheaf categories, and I need some lemma, I often am able to prove the lemma by proving the analogous lemma for sets. As a simple example, let $f_i :X_i\hookrightarrow Y$ be two monomorphisms for $i=1,2$. If $g:X_1 \rightarrow X_2$ be a map such that $f_2\circ g=f_1$ Then $g$ is also a monomorphism. Let us say that this is taking place in simplicial sets (or $\mathcal{A}$-sets). Well, to prove this for simplicial sets, we prove the lemma in sets, and apply pointwise.

What we were able to do is to prove a fact about simplicial sets by reducing to the case of sets. The question is, when can we do this in full generality.