Abstract In this paper we study the holomorphy, with respect to various locally convex topologies, of the composition map ϕ : (f,g) ∈ X × Y → gof E Z of holomorphic functions between Banach spaces E and F, and F and G. If X includes all the constant functions, then ϕ will not be holomorphic when Y is the space of entire functions from F to G. Positive results generally require the topology on Y to be that of uniform convergence on certain subsets of F and the topology on X to be the same type as that on Z, such as compact-open,τ σ , τ ω