One often reads (and writes) that an exact sequence of finite dimensional vector spaces $$ 0 \rightarrow X_1 \rightarrow X_2 \rightarrow \dots \rightarrow X_n \rightarrow 0 $$ induces a canonical isomorphism $$ \bigotimes_{i \; \mathrm{odd}} \Lambda^{\max} (X_ i) \cong \bigotimes_{i \; \mathrm{even}} \Lambda^{\max} (X_i), $$ where $\Lambda^{\max}(X)$ denotes the top exterior power of the vector field $X$. My problem is that there seem to be too many choices for the sign of this ``canonical'' isomorphism. For instance, to the exact sequence $$ 0 \rightarrow X \stackrel{A}{\rightarrow} Y \stackrel{B}{\rightarrow} Z \rightarrow 0 $$ it seems equally canonical to associate the isomorphism $$ \Lambda^{\max}(X) \otimes \Lambda^{\max}(Z) \cong \Lambda^{\max} (Y), \qquad x \otimes B_* (y) \rightarrow A_*(x) \wedge y $$

or the isomorphism $$ x \otimes B_* (y) \rightarrow y \wedge A_*(x). $$

Here $x$ is a generator of $\Lambda^{\max}(X)$ and $y\in \Lambda^{\dim Z}(Y)$ is such that $A_*(x) \wedge y$ generates $\Lambda^{\max}(Y)$.

Since this canonical isomorphism is often used in the theory of determinant bundles in order to define orientations for geometric objects, I find this uncertainty on a sign disturbing.

Reasonable requirements that one should ask to this canonical isomorphism are:

1) to the exact sequence $0\rightarrow X \stackrel{A}{\rightarrow} Y \rightarrow 0$ one associates the isomorphism $x \mapsto A_*(x)$;

2) naturality with respect to isomorphisms of exact sequences.

However, these requirements do not determine the isomorphism uniquely.

My question is: is there a standard convention regarding the definition of the canonical isomorphism which is associated to exact sequences of arbitrary length? And if not, what would be reasonable requirements to add to 1) and 2) in order to have a good definition?