Set-object enriched categories, enriched functors, and enriched natural transformations form a strict 2-category

Take objects to be the set-object $\mathcal{V}$-categories. For fixed $\mathcal{A},\mathcal{B}$, let the objects of the hom-category $\mathcal{V}\text{-}\mathbf{C}\mathbf{a}\mathbf{t}(\mathcal{A},\mathcal{B})$ be the $\mathcal{V}$-functors $\mathcal{A}\to \mathcal{B}$ from [L2], and let the morphisms be the $\mathcal{V}$-natural transformations from [L3]. The identity 2-cell on a functor $T$ has components $j_{TA}:\mathbf{1}\to \mathcal{B}(TA,TA)$ from [L1], and vertical composition of $\alpha :T\Rightarrow S$ and $\beta :S\Rightarrow R$ is defined componentwise by $\mathbf{1}\cong \mathbf{1}\otimes \mathbf{1}\stackrel{{\beta }_A\otimes {\alpha }_A}{\to }\mathcal{B}(SA,RA)\otimes \mathcal{B}(TA,SA)\stackrel{M^{\mathcal{B}}}{\to }\mathcal{B}(TA,RA)$. Associativity and the unit laws for this vertical composition follow directly from the associativity and unit diagrams of [L1].

Horizontal composition of 1-morphisms is ordinary composition of $\mathcal{V}$-functors: if $T:\mathcal{A}\to \mathcal{B}$ and $U:\mathcal{B}\to \mathcal{C}$, then $(UT{)}_{A,B}=U_{TA,TB}\circ T_{A,B}$ and the compatibility axioms follow from those in [L2]. Whiskering of 2-cells is also componentwise: $(U\alpha {)}_A:=U_{TA,SA}\circ {\alpha }_A$ and $(\alpha H{)}_A:={\alpha }_{HA}$. These formulas preserve naturality because [L2] and [L3] are already written against enriched composition.

Step 1.1 gives a category of 1-morphisms and 2-morphisms for every ordered pair of objects, and step 1.2 gives horizontal composition functors between those hom-categories. The associativity and unit laws are strict because they are literal equalities of composed functions and of componentwise composites. Thus [L4] applies.