## Topological limit and categorical limit of a sequence of morphisms

I'm wandering if in a category internal to Top (not just enriched over Top), given a sequence of morphisms that forms a direct (inverse) system, the topological limit of the sequence of morphisms coincides, in some sense (perhaps after quotienting in some way by the equivalence relation induced by the notion "being isomorphic"), with the limit of the system in the categorical sense.

This question (if it makes sense at all), seems quite similar to others already appeared on MO, but I hope it is sufficiently different not to be considered a mere duplicate.

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