Geodesic
Categories of functors between categories with partial morphisms
Discussiones Mathematicae. General Algebra and Applications, Tome 25 (2005) no. 1, pp. 39-87

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It is well-known that the composition of two functors between categories yields a functor again, whenever it exists. The same is true for functors which preserve in a certain sense the structure of symmetric monoidal categories. Considering small symmetric monoidal categories with an additional structure as objects and the structure preserving functors between them as morphisms one obtains different kinds of functor categories, which are even dt-symmetric categories.
Keywords: symmetric monoidal category, dhts-category, Hoehnke category, Hoehnke theory, monoidal functor, d-monoidal functor, dht-symmetric functor, functor composition, cartesian product 
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Vogel, Hans-Jürgen. Categories of functors between categories with partial morphisms. Discussiones Mathematicae. General Algebra and Applications, Tome 25 (2005) no. 1, pp. 39-87. http://geodesic.mathdoc.fr/item/DMGAA_2005_25_1_a2/