If V_ is a symmetric monoidal category but not closed there is stilll a sense in which V_ itself is a V_-category. After all, V_ "acts" on itself via tensor:V_xV_--->V_ and it is useful to think of an arrow X ten Y --->Z as providing an X-indexed family of arrows from Y to Z. If A_ and B_ are categories equipped with actions of V_ one can ask for the relevant notion of functor from A_ to B_ that respects such actions and the condition is the one that you enquire about, your specific case involving endofunctors on the base which having binary products can be seen as a symmetric monoidal category. If the idea of A_ having families of arrows indexed by the objects of V_ is taken as primitive we are led to consider categories A_ equipped with a functor A_^op x V_^op x A_ ---> SET that is suitably compatible with the ordinary hom functor for A_. The relevant conditions ultimately ensure that A_ has the structure of a SET^(V_^op) category. This last is considered to have the monoidal ((bi)-closed) structure introduced by Brian Day. If the data for such is presented as I have it in the last paragraph then the idea is that for A and B in A_ and X in V_ the value of the structure functor at (A,X,B) provides a SET of X-indexed arrows from A to B. This structure admits the possibility of representability in each of the 3 variables so that SET^(V_^op)-category includes the notions of V_-category, "V_-tensored category" and "V_-cotensored category". If A_ and B_ are SET^(V_^op) categories then the notion of SET^(V_^op) functor F between them is clear but there are 3x3=9 special cases that bear inspection in case both A_ and B_ are any of the special cases mentioned above. In each case "the effect of F on homs" (I believe that is the term you want) admits a compact description. In my 1976 thesis there is a table which displays them. Best regards RJ +++++++++++++++++++++++++++++++++++++++++