@NiallTaggart Indeed in the second case, $I(-,k)$ is cofibrant for the projective model structure so Case 2 is also related to homotopy theory and not only to enriched category theory.

Leinster's paper is published in TAC, I have modified the reference.

@DylanWilson The question is badly formulated and badly abstracted from my situation. Consider a coend $\int^{i} X(i)\times D(i)$. What condition should satisfy the diagram $D$ so that by replacing $X$ by a weakly equivalent diagram $Y$, one obtains a weakly homotopy equivalent coend ? (this coend does not have to calculate the homotopy colimit of $X$). I think that something like Theorem 11.5.1 should help.

@DylanWilson I asked the question because I suspect that a coend I have behaves (at least sometimes) as a homotopy colimit and the diagram $D$ does not look at all like the diagram in the question, nor I can see how to realize it as a simplicial set.

I have voted to close this question because I am more than tired by this kind of question. Nobody would never ask such a question by replacing 'category' by anything else you prefer. It is based on prejudices and I would like to share my annoyance here.

@Tyrone I managed to get in touch with him (it turns out that his Google email does not work, the other one works). It seems that the statement is still unproved when the source is not the empty set. You will find the statement (without proof) that $X$ is normal iff $\varnothing\to X$ satisfies the LLP with respect to $g$ in his paper "A diagram chasing formalisation of elementary topological properties" page 7.