# Exponential of monos over terminal object

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

• Yes, $V$ may be arbitrary. Dec 30, 2021 at 12:21
• Thanks. Do you want to make it into an answer? So I can accept it Dec 30, 2021 at 13:08

Yes, $$V$$ may be arbitrary. So long as $$U$$ is subterminal $$U^V$$ will also be subterminal. In other words, you have shown that the class of subterminal objects is an exponential ideal.