Hom-functor preserves pullbacks I am trying to prove that the hom-functor $\mathrm{Hom}(A,-)$ preserves pullbacks.
I stuck at showing uniqueness. Could you please give me some hints?

Consider some pullback $X \times_S Y$ in a category. In order to show that $\hom(A,-)$ preserves it, we have to show that the natural map
$$\hom(A,X \times_S Y) \to \hom(A,X) \times_{\hom(A,S)} \hom(A,Y)$$
is an isomorphism. By construction of pullbacks in $\mathsf{Set}$, the right hand side is simply the set of pairs of maps $A \to X$ and $A \to Y$ such that the diagram
$$\begin{array}{c} A & \rightarrow & X \\ \downarrow && \downarrow \\ Y & \rightarrow & S \end{array}$$
is commutative. Hence, the map is bijective by definition of a pullback!
