I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I state what I learned as follows. Also I will only deal with the $2$-dimensional TQFT here.
$1$. TQFT assigns a $\mathbb{C}$-vector space $Z(M)$ to each $1$-dim closed oriented manifold $M$.
$2$. TQFT assigns a vector $Z(L)\in Z(\partial L)$ to each $2$-dim closed oriented manifold with boundary $L$.
$3$. $Z(M1\cup M2)=Z(M1)\otimes Z(M2), Z(M^*)=Z(M)^*$ ($M^*$ is the orientation reversal of $M$, $Z(M)^*$ is the dual of $Z(M)$)
$4$. For orientation preserving diffeomorphism $\phi : M_1\rightarrow M_2 (M_1,M_2 : \text{1 dim closed manifold})$ there is an induced vector space isomorphism $\phi_{*} : Z(M_1)\rightarrow Z(M_2)$.(plus some naturality)
I omitted other axioms because these seems to be enough for my question. Using 3, we can interpret a vector as a linear map between vector spaces. With this, The theory says that the TQFT assigns Id$_{Z(S^1)}$to the cylinder $S^1\times$I, whose boundary is disjoint union of two $S^1$. Of course they are diffeomorphic, but for later convenience, let's call them $S^1_a $and $S^1_b$. Next I thought as below.
$1$. "every closed oriented $1$-dim manifold is diffeomorphic to each other ($\sim S^1$), even up to orientation preserving diffeomorphism. so there exists isomorphism $Z(S^1_a)\sim Z(S^1_b)\sim Z(S^1)\sim Z(S^{1*})$. But we are not identifying the objects itself via orientation preserving diffeomorphism because both notion $Z(S^1)$ and $Z(S^{1*})$ are appearing in the texts."
$2$. "A choice of the orientation of the cylinder induces orientation($\mu$) on its boundary. since $S^1_a$ and $S^1_b$ are already oriented manifold, comparing this with induced orientation $\mu$ will determine wheter to write each $S^1_a, S^1_b$ as oriented the same way($S^1_i$), or reversely oriented($S^{1*}_i$)."
Because $S^1_a$ and $S^1_b$ are not equal to $S^1$ set theoretically,(yet we do not mod out by diffeomorphism because of "1"?!) so to say this cylinder $(S^1\times I)$ gives a vector in $Z(S^1)\bigotimes Z(S^{1*})$, I thought that I need to choose a diffeomorphism of cylinder and use some naturality axiom to identify $Z(S^1_a)\bigotimes Z(S^1_b)$ with $Z(S^1)\bigotimes Z(S^{1*})$. But I'm not sure this method makes sense since there are lots of (orientation preserving)diffeomorphisms between closed oriented 1 dim manifold. Considering all this, I am also suspicious that why it corresponds to an element of $Z(S^{1*})\bigotimes Z(S^1)$ but not an element of $Z(S^1)\bigotimes Z(S^1)$.
I hope my word was understandable. In short, would anyone help me with the identification problem of $Z(S^1_a)\bigotimes Z(S^1_b)$ and determining its orientation compared to specified $S^1$?
Great Thanks.
I followed the defintion of TQFT in
Atiyah M., Topological quantum field theory , p177-181
Lawrence R.J. , Introduction to topological quantum field theory , p5-7.
Thank you for your attention.