Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$.
This leads to some questions.
1) If $\mathscr{V}$ is (co)complete, is it true that $\mathbf{Cat}_{\mathscr{V}}$ is (co)complete?
Now, let's forget 1) for a moment and get back to the main question. To simplify, let's start by a simple computations (which I'm not sure whether they're correct)
a) The product of two $\mathscr{V}$-enriched categories $\mathscr{C} \times \mathscr{D}$ is given on objects by pairs $(c, d)$ and on morphisms by $\mathscr{C} \times \mathscr{D} ((c, d), (c', d')) = \mathscr{C}(c, c') \otimes \mathscr{D} (d, d')$
b) The coproduct of two $\mathscr{V}$-enriched categories $\mathscr{C} \coprod \mathscr{D}$ is given on objects by the disjoint union and the morphisms are the same ones.
c) The pushout of the category $\mathscr{C} = a \rightarrow b$ with the category $\mathscr{D} = c \rightarrow d$ along the inclusion of the terminal category $*$ (where $0 \in *$ goes to $b$ and $0$ goes to $c$ ) is the category $x \rightarrow y \rightarrow z$ such that $[x, y] = \mathscr{C}(a, b)$ and $[y, z] = \mathscr{D}(c, d)$. Furthermore $[x, z]$ should be intuitively a kind of pushout $\mathscr{C}(a, b) \coprod_X \mathscr{D}(c, d)$ (where $X$ is something that considers the endomorphisms of $b$ and $c$).
So even for the simple example (c) I could not compute the colimit. Clearly, the problem is how to compute the (co)equalizers. Maybe my failure is simply because 1) is not true.
The motivation for this question is that I was trying to compute $\mathfrak{C} [X] = \int^{[n] \in \Delta} X_n \otimes \mathfrak{C}[\Delta^n]$ explicitly for some common $X \in \mathbf{sSet}$, where $\mathfrak{C}[\Delta^n]$ is Cordier enrichment of $[n]$ by declaring $\mathfrak{C}[\Delta^n] (i, j) = (\Delta^1)^{j - i -1}$ (and empty when $i > j$). For instance, I could not compute $\mathfrak{C}[\Delta^2 \coprod_{\Delta^1} \Delta^2]$ (two triangles glued along an edge) and $\mathfrak{C}[\Lambda^n_i]$.
So my question is: what's the general formula for the hom's of a (co)limit of $\mathscr{V}$-enriched categories?
Thanks in advance.