Is it true that there are no nontrivial comonoids (with respect to the cocartesian monoidal structure, of course) in any coslice category of a topos?
Proof that the answer is "Yes" for the case of $Set$
Let $X$ be a set. Then, a comonoid in $X/Set$ consists of a set $Y$ and maps $i:X \to Y$, $p:Y \to X$ (the counit), and $m:Y \to Y \sqcup_{X} Y$ (the comultiplication) satisfying the following conditions:
- $p \circ i=1_X$
- $m \circ i$ coincides with the equal compositions of the two pushout coprojections $Y \to Y \sqcup_{X} Y$ with $i$
- Counitality: The two compositions $Y \to Y \sqcup_{X} Y \to Y$, where the first map is $m$ for both compositions and the second maps are the two maps induced by $i \circ p$ and $1_Y$ (in each order), are both equal to $1_Y$.
- Coassociativity: The two induced ternary comultiplication maps $Y \to Y \sqcup_{X} Y \sqcup_{X} Y$, namely $(m \sqcup_{X} 1_Y) \circ m$ and $(1_Y \sqcup_{X} m) \circ m$, are equal.
Now, one must prove that $i$ and $p$ must in fact be inverse bijections. In fact, only the first three conditions are needed; coassociativity is not used at all.
First, note that $Y \sqcup_X Y$ (the "amalgamated coproduct") can be seen as the quotient of $Y \times \{0,1\}$ by the equivalence relation that identifies $(y, a)$ with $(y', a')$ if $y=y'$, and either $a=a'$ or $y$ is in the image of $i$. Given this, for any $y \in Y$, $m(y)$ is an equivalence class $[(y', a)]$ where $y' \in Y$ and $a \in \{0,1\}$.
Then, one of the two maps $Y \sqcup_{X} Y \to Y$ sends $[(y', a)]$ to $i(p(y'))$ if $a=0$ and $y'$ if $a=1$, and the other map sends $[(y', a)]$ to $y'$ if $a=0$ and $i(p(y'))$ if $a=1$. Regardless of whether $a$ is $0$ or $1$, one of those two maps must send $[(y', a)]$ to $i(p(y'))$, and counitality then implies that one must have $y=i(p(y'))$.
Hence, $i \circ p$ is surjective. But it is also idempotent, so it must in fact be equal to $1_Y$. Hence, $i$ and $p$ must in fact be inverse bijections. $\square$
General case
Since pushouts in presheaf toposes are computed pointwise, the answer to the question continues to be "Yes" for any presheaf topos (even if one replaces "comonoid" with "counital comagma"). But I do not see how the above argument could internalize to any (elementary) topos.