I hydrolized the following compound: cis-norbornene-5,6-endo-dicarboxylic anhydride

then took the cis-norborene-5,6-endo-dicarboxylic acid and added H2SO4 to it (warmed it up and cooled it) then added drop by drop water (then I reheated the mixture and crystallized it again).

What is the compound? It was positive for carboxylic acid and also positive for an ester.

The structure of my compound can be found on google images "norbornene dicarboxylic acid"

It's a 2 five member ring (one with C=C bond) and it's connected via triangle of c-c bonds, with CO2H popping out from 2nd ring without C=C bond

When I was looking for answer online for what the compound was someone said that it was "add water across the double bond, open the anhydride, and then form a lactone with the carboxylic acid that is closer to the hydroxide" but I'm still not sure what the structure is and if they are actually correct. I was wondering if it's possible to explain it with arrow pushing mechanism please

Based on your description of the reactions you performed, it seems that the initial compound, cis-norbornene-5,6-endo-dicarboxylic anhydride, was first hydrolyzed to form cis-norbornene-5,6-endo-dicarboxylic acid. Then, by adding H2SO4 and heating followed by dropwise addition of water and reheating, you likely carried out an intramolecular reaction involving the carboxylic acid and the double bond.

Here's a possible mechanism for the reaction, as you requested:

1. Start with cis-norbornene-5,6-endo-dicarboxylic acid:
HOOC-C(H)=C(H)-C(H)(COOH)-C(H)(COOH)

2. Add H2SO4 and heat:
The sulfuric acid will protonate the carboxylic acid group making it a better leaving group. This can lead to the electrophilic attack of the double bond by the carboxylic acid.

3. The carboxylic acid attacks the double bond:
The pi electrons of the double bond act as a nucleophile and attack the carbonyl carbon of the carboxylic acid group. Simultaneously, the electrons in the carbonyl oxygen of the carboxylic acid leave, creating a new double bond. This results in the formation of a cyclic structure called a lactone.

4. Reheat and add water dropwise:
By reheating and adding water, you likely performed hydrolysis of the lactone ring back to the carboxylic acid. The water molecule acts as a nucleophile and attacks the carbonyl carbon of the lactone, breaking the bond and restoring the carboxylic acid functionality.

The overall reaction can be summarized as follows:

cis-norbornene-5,6-endo-dicarboxylic acid (starting material) -> intramolecular reaction -> lactone intermediate -> hydrolysis -> final compound (carboxylic acid)

It's important to note that without specific information about the conditions and reagents used in each step, it's difficult to provide a definitive answer. However, based on your observations of positive tests for carboxylic acid and ester, it's likely that the reaction proceeded as described above.