Consider a cartesian closed category $\mathcal E$. Let $1$ be the terminal object and $U, V\hookrightarrow 1$ to monomorphisms into $1$. I wanted to prove that $U^V\rightarrow 1$ is also a monomorphism. So, we want to prove that for every $X$ there is at most one map $X\rightarrow U^V$. This should be immediate maps $X\rightarrow U^V$ are in 1-1, onto correspondence with maps $X\times V\rightarrow U$, but $U\rightarrow 1$ is mono, so there is at most one map $X\times V\rightarrow U$.
Now, I didn't use that $V\rightarrow 1$ is mono, so I would conclude that this hypothesis is superfluous. Is my conclusion correct? For reference, I'm working out the statement in "Sheaves in Geometry and Logic", page 201.
The exponential of $U^V$ of two open objects $U\hookrightarrow 1$ and $V\hookrightarrow 1$ is again open, i.e., $U^V\rightarrow 1$ is monic
