Basic Combinatorics Question from Article 1 of Lawvere's Conceptual Mathematics

Question

the question I am referring to is: "Can you find a pair of maps $B \stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} B$ for which $g \circ f = 1_B$?" Where $A$ is a set of 3 distinct elements and $B$ is a set of 2 distinct elements. We are then asked to count the number of pairs of maps satisfying the above condition. This seems like a very basic question to me as there are 6 valid maps for $g$, each of which may pair up with 2 options for $f$ giving 12 total pairs. However, online I have seen a number of sources give 6 as the answer to this question. Am I missing something obvious or are those other answers incorrect?

Answer 1

Looks to me that you are correct.

You first identified that there are $6$ possible surjective functions $g$. I will go the other way around, there are $3\cdot 2=6$ possible injections $f$. The image of $f$ is of cardinality $2$, and $g$ is uniquely determined on those two elements by $g\circ f = 1_B$. The remaining element of $A$ not in the image of $f$ is free to go wherever it wants, and there are two possible choices. This gives us total of two choices of $g$ for each $f$, which indeed gives total of $12$ pairs.