ct.category theory - Does linearization of categories reflect isomorphism?

In this answere I (try to) present the problem as a Algebraic Geometry one:

consider the category $mathscr{C}$ with two objects $X, Y$ and

$mathscr{C}(X, Y)$={$r_1, s'_1, r_2, s'_2$} ;
$mathscr{C}(Y, X)$={$s_1, r'_1, s_2, r'_2$} ; $mathscr{C}(X, X)$={$1_X, e_X$} ; $mathscr{C}(Y, Y)$={$1_Y, e_Y$} where $e_X, e_Y$ are idempotent, and any composition of a morphism by a a idempotent not alter the morphism, and $1_Y= r_1circ s_1= r_2circ s_2$, $1_X= r'_1circ s'_1= r'_2circ s'_2$, all other compositions give the (no identity) idempotent.
Suppose that $R$ is a commutative ring and in $Rmathscr{C}$ consider the morphims
$A:= a_1cdot r_1 + b'_1cdot s'_1 + a_2cdot r_2 + b'_2cdot s'_2: Xto Y$ and

$B:= b_1cdot s_1 + a'_1cdot r'_1 + b_2cdot s_2 + a'_2cdot r'_2: Yto X$.

Let $alpha :=a_1+b'_1+ a_2+ b'_2$, $beta :=b_1+a'_1+ b_2+ a'_2$,

Then we have $Bcirc A=1_X$ iff:

1) $a'_1cdot b'_1+ a'_2cdot b'_2=1$ and

2) $beta cdot a_1+ (beta - a'_1)cdot b'_1+ beta cdot a_2+(beta -a'_2)b'_2=0$ i.e. $beta cdot alpha = a'_1cdot b'_1+ a'_2cdot b'_2$

similarly we have $Acirc B=1_Y$ iff:

1') $a_1cdot b_1+ a_2cdot b_2=1$ and

2') $alpha cdot beta = b_1cdot a_1+ b_2cdot a_2$

all these equations are equivalent to the system of three equations:

$a'_1cdot b'_1+ a'_2cdot b'_2=1, a_1cdot b_1+ a_2cdot b_2=1, alpha cdot beta = 1$

thinking these in $mathbb{C}[ a_1, b'_1, a_2, b'_2 , b_1, a'_1, b_2,a'_2]$ these represent three varieties on $mathbb{C}^8$

If these varieties have an a intersections then $X, Y$ are isomorphic in $mathbb{C}mathscr{C}$ (but aren't isomorphic in $mathscr{C})$.

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