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The Graded Classification Conjecture states that the pointed$K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt pathalgebras of finite graphs when these algebras are considered with their naturalgrading by $\mathbb Z.$ The strong version of this conjecture states that thefunctor $K_0^{\operatorname{gr}}$ is full and faithful when considered on thecategory of Leavitt path algebras of finite graphs and their gradedhomomorphisms modulo conjugations by invertible elements of the zerocomponents. We show that the functor $K_0^{\operatorname{gr}}$ is full for theunital Leavitt path algebras of countable graphs and that it is faithful(modulo specified conjugations) only in a certain weaker sense.

The functor $K_0^{\operatorname{gr}}$ is full and only weakly faithful