Exercise about map objects $Y^{1}$ ("Conceptual Mathematics Second Edition", p.314, Article V Map objects, Exercise 3)

Question

Suppose $\mathcal C$ has products and a terminal object $1$. If the map object $Y^{1}$ exists, I'm having trouble proving that $Y^1 \simeq Y$. I thought this would pop-out easily as follows: Let $p: 1 \times Y \rightarrow Y$ denote the right projection. Then the UMP of map objects provides a unique map $\ulcorner p \urcorner : Y \rightarrow Y^{1}$ such that $p = e \circ (1_{1} \times \ulcorner p \urcorner)$. Here $e$ is the evaluation map $1 \times Y^{1} \rightarrow Y$ in Lawvere's notation. Since $1$ is a terminal object, it follows that $p$ is an isomorphism, and hence has both a left and right inverse. I hoped from here to conclude that $\ulcorner p \urcorner$ is also an isomorphism, but I end up going around in circles! Can anyone help?