Definition Pseudomorphism of pseudocategories [efr-8YZ9]

Let C = (C_0,C_1, d,c,e,m,\alpha ,\lambda ,\rho ) D = (D_0,D_1, d',c',e',m',\alpha ',\lambda ',\rho ') be internal pseudocategories in a 2-category \mathbb {C}. A pseudomorphism F: C \to D consists of the following data:

1. Morphisms F_0: C_0 \to D_0, F_1:C_1 \to D_1 in \mathbb {C}
2. Isomorphism 2-cells \mu : F_1m \to m(F_1 \times _{F_0} F_1) and \epsilon : F_1e \to eF_0

So that the following equations hold:

1. d'F_1 = F_0d, c'F_1 = F_0c
2. d' \circ \mu = 1_{F_0}d\pi _2, c' \circ \mu = 1_{F_0c\pi _1}
3. d' \circ \epsilon = 1_{F_0}, c'\circ \epsilon = 1_{F_0}

And so that the following diagrams commute: