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What is the right notion of a functor from an internal topological category to a topologically enriched category?

I don't believe it is possible to recover the "correct" notion of "functor $\mathcal{C}\to \rm Top$", as described at the other question you linked to, by viewing $\rm Top$ only as ...
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10 votes
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Proposition A.2.6.15 in HTT

Retracts of weak equivalences are weak equivalences. Now if $f'$ is a retract of $f$ and you start with such a diagram with $f'$ on the left, you can create a new diagram with $f$, the same $X', X''$ ...
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5 votes
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Fibrant replacement of an injective model category of enriched diagrams

Section 8 of my paper All (∞,1)-toposes have strict univalent universes shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of ...
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