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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1092,
author = {Hans-J\"urgen Vogel},
title = {Categories of functors between categories with partial morphisms},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {25},
year = {2005},
pages = {39-87},
zbl = {1094.18004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1092}
}
Hans-Jürgen Vogel. Categories of functors between categories with partial morphisms. Discussiones Mathematicae - General Algebra and Applications, Tome 25 (2005) pp. 39-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1092/
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