Conceptual definition: Injection, surjection and Bijection. I was wondering if this conceptualisation is correct:
Injection means that we don't have two arrows come from an element of the domain towards the range. Hence we don't have one archer standing in the domain, shooting two targets by himself.
Surjection means that all of the targets have been hit by these archers.
Bijection means that each archer standing on the domain, shoots one arrow each into the range, hitting only one target each hitting all of the targets.
Is this conceptualisation correct?

Updated attempt:
Injection means that we dont have two archers standing on the domain, hitting the same target. This, meaning, that there is no competition among archers.
Bijection means that these successful archers do not compete and each fires a single arrow, successfully defeating every target. Hence one arrow each is fired each hitting one unique target.
 A: In the way you are telling,let us define a function from collection of all archers to the collection of targets by,
function : each archer hits exactly one target.

Then we can make the following definitions:
injective function : function in which no two archers will hit the same target
surjective function : function in which each target has been hitted by atleast one archer, i.e., no target remains without hitting.
bijective function : function in which each target has been hitted by exactly one archer.
A: Maybe a picture could be helpful?
Loosely speaking, these concepts try to formalize some kind of "size" relation between the two sets. In your terms of archers and targets, if you have an injective function you can think of a situation where the targets are at least the same number as the archers. In the case of a surjective function the targets are at most equal to the number of archers and finally if you have a bijective function you have the same number of archers and targets.
(side note: to make a bijective funtion out of an injective one, you can just discard some of the targets - you cannot do this if you start with a surjective function which is not injective).
Again, this is an extremely loose intuitive explanation, hope it helps.

