In section 6.5 of Basic Concepts of Enriched Category Theory Kelly writes
If $\lambda : F \to \mathcal A(T-,M)$ is a cylinder in $\mathcal A$, where $F : \mathcal L^{op} \to \mathcal V$ and $T : \mathcal L \to \mathcal A$ we consider for $B \in \mathcal B$ the functor
$$ \mathcal L \cong \mathcal L \otimes \mathcal I \xrightarrow{T\otimes B} \mathcal A \otimes\mathcal B $$
sending $L$ to $(TL,B)$, and called $T\otimes B$. Then we can consider the cylinder
$$ F \xrightarrow \lambda \mathcal A(T-,M) \cong A(T-,M)\otimes I \xrightarrow{1\otimes j} A(T-,M)\otimes \mathcal B(B,B) = (\mathcal A\otimes\mathcal B)((T-,B),(M,B)) $$
and call this $\lambda\otimes B$. Note that this is a colimit cylinder in $\mathcal A\otimes \mathcal B$ if $\lambda$ is one in $\mathcal A$; essentially because $- \otimes Y : \mathcal V \to \mathcal V$ preserves colimits.
Now I'm a bit confused, because I think for ordinary categories and conical limits, ie. $\mathcal V = \mathcal{Set}$ and $F = \Delta_1$, $\lambda \otimes B$ would only be a colimiting cone if the indexing category $\mathcal L$ is connected. If we take coproducts, ie. $\mathcal L$ is the discrete category with two elements $L_1, L_2$, $\lambda \otimes B = \lambda \times B$ is the cone from the base $(TL_i,B)$ to $(\coprod TL_i,B)$. But $\coprod (TL_i, B)$ is $(\coprod TL_i, B^{(2)})$.
Am I getting something wrong or is this a mistake in the book? I suppose if it was, it may still be fine as we later pass to the $\mathcal F$-theory in which these go to colimiting cylinders anyway by construction.