General question:
When is a coslice of a cartesian closed category cartesian closed?
While slices of cartesian closed categories are often themselves cartesian closed, coslices are rarely cartesian closed.
In fact, we have the following theorem (with its corollary):
Theorem: Given an object $Q$ of a cartesian closed category $C$ with finite coproducts, the coslice category $Q/C$ is cartesian closed if and only if the unique map from the initial object $0$ to $Q$ is epic.
Proof: Suppose that the unique map $0 \to Q$ is epic. Then, $Q/C$ can be identified with the full subcategory of $C$ consisting of objects having a (necessarily unique) map from $Q$. This means that $Q/C$ will in fact be an exponential ideal closed under finite products in $C$ (if there is a map $Q \to Y$, then there will also be a map $Q \to Y^X$ for any object $X$, regardless of whether there is a map $Q \to X$), so is in particular cartesian closed.
Conversely, suppose that $Q/C$ is cartesian closed. Then, the codiagonal map $\nabla_X:(Q+Q, i) \to (Q, 1_Q)$, where $i$ is one of the two coproduct injections, is a map with initial codomain in $Q/C$, and hence is an isomorphism because initial objects in cartesian closed categories are strict. This implies that the unique map $0 \to Q$ is epic. $\square$
Corollary: If the unique map $0 \to Q$ is epic, then also the unique map $Q \to 1$ is monic, where $0$ and $1$ are respectively the initial and terminal objects of $C$. $\square$
The above theorem, however, only answers the question in the special case where the cartesian closed category has finite coproducts.
Pathological examples of cartesian closed categories $C$ with an object $Q$ such that the coslice category $Q/C$ is cartesian closed but there is at least one object of $C$ that has more than one map from $Q$ may or may not turn out to exist.