It is a common belief that the con- prefix means “together” or “with”.
I think in certain cases, the con- prefix can also act as an “interior-izar”. It turns an otherwise exterior verb, acting on the outside world, to an interior verb acting on itself, and might even symbolise the spiritual or emotional understanding of that person.
Unfortunately I couldn’t find support (from other people) to my theory about this role of the con- prefix. But here are some word-proofs that I found in Spanish (which is a language I’m learning at the moment).
My best argument is the parallelism between the verb tener (to hold) and contener (to contain, i.e. to hold within).
Another good example is the verb mover (to move) and conmover (to cause emotional movement, i.e. to “touch”, as in this piece of art touched me, or the poem touched me).
Vencer (to overcome, to defeat) and convencer (to persuade) can also support my theory. Many languages see the connection between overcoming and persuading. If con- acts as an interior-izer it can mean to overcome from within. I.e. to defeat your opponent from within. But the together explanation can also work (though I think in a lesser degree), as it can mean to overcome the enemy and bring him to be with- you.
Another example is the verb vivir (to live) and the verb convivir, which means to co-exist, and is often thought of as “to live together”. But I think this is a mistake. To translate co-exist as to live together is redundant (i.e. the sentence “A vivir con B” and “A convivir con B” has the same meaning if convivir means “to live together”). Co-exist does not refer to the outerior act of living together. People can live together without coexistence (as my country of Israel can testify for). It is only when people inner feelings, emotions and beliefs can “live together”, that they can truly co-exist, or convivir.
Convenir is a tougher verb. It is of the same root as the English word convention. It has two translations - A “to be a good idea”, and B “to be agreed upon”. Venir means to come. If con- means together, it will mean to come-together. This makes good sense in the form of everyone came together and decided upon X, i.e. for meaning B, but not so much for meaning A. Meaning A - it is not being used as a good idea by convention, but rather as someone’s own private notion of what is a good idea. Here it can make more sense if con- means “from within”. As in - It comes to me from within that you should do x or y or z. If convenir means to come from within, as in a strong intuition, or belief or creed about what one should do - it makes more sense for meaning A.