Rewording this problem via what Zhen Lin's notion of the original question is.
Let $X$ and $Y$ be ringed spaces. Prove that, for each open $U \subset X$, the presheaf $$U \mapsto \mathrm{Hom}(U, Y)$$ is a sheaf.
Is this because we can always get a presheaf (in fact, a sheaf) on any topological space $X$ by fixing some topological space $Y$ and sending any open subset $U$ of $X$ to the set of continuous functions from $U$ to $Y$ with the obvious restriction maps?
How would I go about proving this?
Yes, you're just asked to show that if you have continuous functions $f_i$ on $U_i$ agreeing on $U_i\cap U_j$ then they extend to a unique continuous function $f$ on $\cup U_i$, which is true because continuous functions are defined pointwise and continuity is local.
Edit: You also need to show that the homomorphisms $f_i^\#:\mathcal{O}_Y\to(f_i)_*\mathcal{O}_X|_{U_i}$ glue. But since for some $\sigma\in\mathcal{O}_Y(V)$ $f_i^\#\sigma$ and $f_j^\#\sigma$ agree on the common pushforward $(f_i)_*\mathcal O_X|_{U_i\cap U_j}=(f_j)_*\mathcal O_X|_{U_i\cap U_j}$, we have a coherent system of sections of $\mathcal{O}_X(f^{-1}(V))$ over the cover $f^{-1}(V)\cap U_i$ which glue uniquely to $f^\#\sigma\in f_*\mathcal{O}_X(V)$.