Unitalities and mapping spaces in $A_{\mathrm{\infty }}$-categories

We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify various models for internal homs and mapping spaces in the infinity-categories of dg-categories and of A-infinity–categories, generalizing results of Toën and Faonte.