CatDat

Implication Details

Claim: If a category is cartesian closed and has a strict terminal object, then it is thin.

Proof: If a morphism $X \to Y$ exists, we get a morphism $1 \to [X,Y]$, which forces $[X,Y]$ to be a terminal object by assumption. But then any two morphisms $1 \rightrightarrows [X,Y]$ are equal, so that any two morphisms $X \rightrightarrows Y$ are equal.