I am reading the paper by Kuznetsov and Lunts, *Categorical resolutions of irrational singularities*, and I’m struggling with a few things. The definition of gluing of DG-categories $\mathcal{D}_1$ and $\mathcal{D}_2$ along a bimodule $\phi \in \mathcal{D}_2^{op} \otimes \mathcal{D}_1$ is the following: they say that an element of such a category is a triple $(M_1, M_2, \mu)$, with $M_i \in \mathcal{D}_i$ and $\mu \in \phi(M_2,M_1)$ a closed element of degree zero and then they define the morphisms of degree $k$ between two such triples to be
$$
Hom^k(M_1,N_1) \oplus Hom^k(M_2,N_2) \oplus \phi^{k-1}(N_2,M_1).
$$
My questions are related to this definition:

1) Can we change the definition of the morphisms and instead of $\phi^{k-1}(N_2,M_1)$ put $\phi^{k-1}(M_2,N_1)$ (eventually changing the definition of the differentials and the composition law)?

2) The differential is defined in the obvious way in the first two components, while in the third is defined as $$ -d(f_{21}) - f_2 \circ \mu + \nu \circ f_1 $$ where the map we are considering is $(f_1,f_2,f_{21})$ and the objects are $(M_1,M_2,\mu)$, $(N_1,N_2,\nu)$. I don’t understand what’s the meaning of the last two pieces, how should I intend the compisition? Thanks in advance.

**EDIT:** I understood the answer to question 2, my mistake was that I wasn't taking into account that we were talking about right modules. Still, I'd like to know the answer to question 1.