Category Theory: homset preserves limits
page 9 of http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Henderson.pdf
Let $D : \mathcal J \to \mathcal C$ be a diagram for some locally small category $\mathcal C$.
Let $(L,\forall i \in \mathcal J, L \overset{\lambda_i}{\to} D(i))$ be a limit cone of that diagram.
Let $X$ be any object of $\mathcal C$.
Let $(S,\forall i \in \mathcal J, S \overset{f_i}{\to} \mathcal C(X,D(i)))$ be any cone over the diagram $\mathcal C(X,D-) : \mathcal J \to \mathbf{Set}$.
We want to construct a unique map $u : S \to \mathcal C(X,L)$ that makes the legs commute.
Given $s \in S$, we have a cone $(X,\forall i \in \mathcal J, X \overset{f_i(s)}{\to} D(i))$ in $\mathcal C$ and thus a universal map $u_s : X \to L$ such that $f_i(s) = \lambda_i \circ u_s$ for each leg, and the triangles of the cone commute because $f_i(s)$ came from a cone.
This gives us a map in set $u : S \to \mathcal C(X,L)$ that makes all the legs of the cone commute. It is also the unique such map because it is pointwise unique.
That proves the theorem.
There is a rather silk proof which requires some observations:
Let $F$ be a diagram in $\mathcal{C}$ indexed by $\mathscr{J}$. Then, a limiting cone $(\lim F, \mu\colon \Delta \lim F \to F)$ corresponds exactly to an isomorphism $$ \mathcal{C}(X, \lim F) \cong [\mathscr{J}, \mathcal{C}](\Delta X, F). $$ natural in $X$ where the right hand side is just the set of cones over $F$.
The set of $F$-cones with tip $B$ is isomorphic to $[\mathcal{J}, \mathbf{Set}](\Delta 1, \mathcal{C}(B, F-))$.
When $\mathcal{C} = \mathbf{Set}$ and $X = 1$ in the first isomorphism, we have $$ \lim F \cong [\mathcal{J}, \mathbf{Set}](\Delta 1, F) $$ i.e. the set of cones from a singleton set $1$ over $F$ is the limit of $F$.
Finally, we arrive the conclusion that the set $\mathcal{C}(B, \lim F)$ is the limit of $\mathcal{C}(B, -) \circ F$ from $$ \mathcal{C}(B, \lim F) \cong [\mathcal{J}, \mathcal{C}](\Delta B, F) \cong [\mathcal{J}, \mathbf{Set}](\Delta 1, \mathcal{C}(B, F-)) \cong \lim \mathcal{C}(B, F-) $$ where the first isomorphism follows from our first observation, the second from the second observation, and the third from the inverse of the third observation.