An internal hom for Conway addition

## Abstract In Games as recursive coalgebras the category Game of impartial combinatorial games is shown to be symmetric monoidal closed for the Conway addition, but only by an abstract adjoint functor argument; Question 5.7 of that paper (Problem 5.0.5 of the author's problem list) asks what the internal hom actually is. We answer this. For games G, H we exhibit [G, H]₊ as the largest subgame A(G, H) of the cofree game on the set Set(UG, UH) on which evaluation of labels is a game morphism. Concretely, a position of [G, H]₊ is a finite well-founded tree whose nodes are labelled by functions UG → UH, subject to a hereditary option equation, and a move is the passage to a child. We prove that evaluation is a game morphism, that A(G, H) is maximal with this property, that currying is a bijection natural in all three variables, and we compute several examples. In particular [∗1, ∗n]₊ ≅ 1 for every n ≥ 1 and [∗2, ∗1]₊ = ∅, so this internal hom is emphatically not the −G + H of Joyal's compact closed category of games and strategies. The relation between the two remains open.