Missing cogenerating collections

Lemma.

Let $\C$ be a category and $U: \C \to \Set$ be a functor. Assume there exists a collection of objects $\F \subseteq \Ob(\C)$ satisfying the following conditions:

For any $X \in \F$ and any non-terminal $Y \in \C$ for every morphism $f: X \to Y$ its underlying map $U(f) : U(X) \to U(Y)$ is injective.
For every infinite cardinal number $\kappa$ there exists an object $X \in \F$ such that $\card(U(X)) \geq \kappa$ and such that $X$ has a non-identity endomorphism.

Then $\C$ does not have a cogenerating collection.

Proof. Assume that there is a cogenerating collection $S$ In particular, $S$ is essentially small. By assumption (2) applied to a cardinal $\kappa > \sup \{\card(U(Y)) : Y \in S\}$ there is an object $X \in \F$ such that $\card(U(X)) > \card(U(Y))$ for all $Y \in S$ and which has a non-identity endomorphism $\sigma : X \to X$ Since $S$ cogenerates, there is a morphism $f : X \to Y$ with $Y \in S$ and $f \sigma \neq f$ For this, $Y$ must be non-terminal. By (1) the map $U(f) : U(X) \to U(Y)$ is injective. This is a contradiction. $\square$

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