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Let's quote the definition of a weak limit for a diagram in a category $C$ that we can find in nLab:

A weak limit for a diagram in a category is a cone over that diagram which satisfies the existence property of a limit but not necessarily the uniqueness.

Let $C$ be a category that has weak limits for a diagram $D: I \to C$. Suppose that $C$ also has limits for $D$, are then all weak limits also limits? If this is not the case, can you provide a counterexample of a category that has both proper weak limits and limits for the same diagram $D$?

I tried to prove that a category with limits for a diagram $D$ cannot have proper weak limits for the same diagram: Let $W$ be a weak limit over $D$. As $C$ also has proper limits for $D$, let $L$ be such a limit cone. But then there must be a unique morphism of cones $\phi: W \to L$, and, by the existence property, a (not necessarily unique) morphism of cones $\psi: L \to W$. By the finality of $L$, we have that $\phi \circ \psi = \mathsf{id}_L$. However, I didn't manage to prove the other half of the isomorphism, namely $\psi \circ \phi = \mathsf{id}_W$.

In case a counterexample can be found, is it possible to find conditions on the diagram $D$ that ensure that weak limits and limits on $D$ coincide if they exist?

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Actually, it is easy to see that every inhabited set is a weak terminal object in the category of sets, but only the singleton sets are terminal objects. Hence, there are proper weak terminal objects in the category of sets.

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