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Prove that if $X^X$ is a terminal object then $X \to \mathbf{1}$ is a monomorphism

I want to know how to solve the following exercise in the textbook Conceptual Mathematics by Lawvere [Session 31, Exercise 2].

> Let $X$ be an object in a cartesian closed category. Show that the following
> two properties are equivalent:
> 
> 1) $X \to \mathbf{1}$ is a monomorphism;
> 2) $X^X = \mathbf{1}$.

I see that (1) is the same as saying that for all objects $A$, there is at most one map $A\to X$. Using this, I can easily prove that (1) implies (2), but the other direction eludes me.